Inverse Complete Elliptic Integrals of First and Second type.

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To implement with the elliptic package the inverse Complete Elliptic Integrals 
of First K(k) and Second Type E(e).

Various hints from: Louis V. King On The Direct Numerical Calculation Of 
Elliptic Functions And Integrals, 
http://www.archive.org/details/onthenumerical032686mbp

Original issue reported on code.google.com by moiseev....@gmail.com on 20 Dec 2010 at 6:45

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Some discussions on the topic
http://mathoverflow.net/questions/48468/invert-complete-elliptic-integral-of-fir
st-kind-kk

Original comment by moiseev....@gmail.com on 20 Dec 2010 at 7:10

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For the procedure can be used the Lagrange Inversion Theorem
http://en.wikipedia.org/wiki/Lagrange_inversion_theorem

Original comment by moiseev....@gmail.com on 19 Jan 2011 at 12:28

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Topic related discussion: 
http://www.mathkb.com/Uwe/Forum.aspx/math/64111/Inverse-of-complete-elliptic-int
egrals

Original comment by moiseev....@gmail.com on 19 Jan 2011 at 12:29

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OK, it seems it is possible to accomplish using the inversion of series 
expansion described here: http://people.math.sfu.ca/~cbm/aands/page_16.htm

And the series for E
Emphi := phi 
         -1/6*m*phi^3 
         +1/5*(1/6*m-1/8*m^2)*phi^5 
         +1/7*(-1/45*m+1/12*m^2-1/16*m^3)*phi^7 
         +1/9*(1/630*m-1/40*m^2+1/16*m^3-5/128*m^4)*phi^9 ... 

Inverts into 
phi := E(phi,m) 
       +1/6*m*E(phi,m)^3 
       +1/120*m*(13*m-4)*E(phi,m)^5 
       +1/5040*m*(493*m^2-284*m+16)*E(phi,m)^7 
       +1/362880*m*(37369*m^3-31224*m^2+4944*m-64)*E(phi,m)^9 ... 

The discussion may be find here: 
http://groups.google.com/group/sci.math.research/browse_frm/thread/6eb1aa7e07b1c
382

Original comment by moiseev....@gmail.com on 6 Oct 2011 at 11:02

[moiseevigor]

Partially closed by INVERSELLIPTIC2